I greatly enjoy this discussion of multiplication. For me theeye-opening moment was Liping Ma's question# for "the other"word problem for 1 3/4 divided by 1/2. It did not take long,but I realized that I was not fit (w/o further training) toteach K-4 math. As I quickly realized only after her question,clearly there are at least two different word problems, askingfor either of the two factors in a * b = c. Here a = c/b andb = c/a, but a and b have different stories (e.g. number ofpiles and number of candies in each pile, or, weight/size ofeach loaf / original loaf and number of loaves, or, as I likebetter, the magnification factor or the size).

When starting to use CAS in the classroom almost 20 years ago,I first was upset, then loved the idea that the graphs ofy=x^2 and y=3x^2 looked exactly the same --- not what we weresupposed to do according to the syllabus, but a view of theworld that might be even more useful (we need both). The factor3 may be seen just as the scaling, choice of units, the physicalobject/relation ship is unchanged ...